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Empirical Economics

03.05.2024

Point forecasts of the price of crude oil: an attempt to “beat” the end-of-month random-walk benchmark

verfasst von: Nima Nonejad

Erschienen in: Empirical Economics

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Abstract

The study of Ellwanger and Snudden (J Bank Financ 154:106962, 2023) discovers a new and remarkable finding regarding the ability of the random-walk model using the end-of-month price of crude oil to forecast future monthly average crude oil prices out-of-sample. The magnitude and nature of the relative predictive gains lead the authors to question whether any other model can “beat” the end-of-month price random-walk out-of-sample. I make an attempt to do so by relying on plain end-of-month crude oil price autoregressive fractionally integrated moving average (ARFIMA) models. These models are more nuanced and at the same time comprehensively account for one of the most salient features of the price of crude oil, namely, its persistence. Consequently, a forecaster is inclined to believe that they might “beat” the end-of-month random-walk model. However, out-of-sample results demonstrate that a uniform (definitive) conclusion cannot be drawn. On the contrary, conclusions depend heavily on the definition of “beating”, i.e. population-level versus finite-sample relative predictability, the forecast horizon, state of the business cycle and the choice of the crude oil price series itself. The decisions, judgments and dilemmas faced by the forecaster are presented and elaborated.

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Here, one cannot proceed without mentioning the significant contributions of Hamilton (1983, 1996, 2008) and Hamilton (2009) with regard to establishing the connections between crude oil price increases and recessions in the USA.

Under the null hypothesis that the end-of-month and monthly average crude oil prices both follow a random-walk, the authors show that the percentage improvement in the mean square error ratio from using the end-of-month random-walk instead of the monthly average counterpart has a closed-form solution and is a function of the forecast horizon, and the number of business days in the month.

A reader not familiar with fractionally integrated (long-memory) models, can interpret them as essentially autoregressive (AR) or moving average (MA) models of a very high order, say m, where \(m\thickapprox T\), and T is the sample size. However, contrary to the plain AR (MA) model of order m, where there are m autoregressive (moving average) coefficients to be estimated, the coefficients are a function of only one parameter, namely, d, which is determined via the Gamma function, see Sect. 2 for details.

As argued in studies, such as Tang and Xiong (2012) and Kartsakli and Adams (2020) among others, a consequence of financialization of crude oil is that the price of crude oil increasingly co-moves with equity markets. Therefore, variables that are used to forecast returns on equities out-of-sample can also be conditioned on to forecast the price of crude oil out-of-sample. Likewise, based on results reported in studies, such as Degiannakis et al. (2018) and Yang (2019), researchers justify conditioning on newspaper-based uncertainty measures, such as economic policy uncertainty and geopolitical risk to forecast returns on the price of crude oil by arguing that these measures capture information regarding future crude oil supply (demand) disruptions.

I can also refer the reader to the seminal studies of Granger (1980); Granger and Joyeux (1980) and Hosking (1981) for an introduction to the concept of fractional integration.

Hosking (1981) shows that a covariance stationary ARFIMA process admits an infinite autoregressive (moving average) expression. In other words, under the assumption of covariance stationarity, (2.1) can be expressed as \(y_{t}=\mu +\Sigma _{j=1}^{\infty }\pi _{j}y_{t-j}+\eta _{t}\). As displayed by (2.2), the method of Beran (1994) consists of simply replacing the infinite sum by a finite one.

The forecaster (econometrician) does not a priori know the number of regimes (breaks) in the data generating process. Therefore, she (he) must estimate the model conditional on 0, 1, ..., m regimes (breaks) and determine the optimal number of regimes (breaks) using a criterion, such as DIC or the marginal likelihood, see for example, Nonejad (2015) and Nonejad (2019).

Results reported in Raggi and Bordignon (2012); Nonejad (2015) and Nonejad (2019) indicate that relative out-of-sample gains obtained by accounting parameter instability have more to do with density forecasts rather than point forecasts.

To compute (2.3), the adjusted out-of-sample squared point forecast error differences are regressed on a constant. Then, the t-test is performed using a heteroscedasticity and autocorrelation-consistent (HAC) variance.

The Clark and West (2007) testing framework assumes that point forecasts are produced under linear regressions estimated via OLS. Therefore, in Sect. 4.3, a Monte Carlo analysis is conducted to explore the properties of the test under models, such as (2.1). It is concluded that the Clark and West (2007) test has strong power and is well-sized under data generating parameters.

Results from Monte Carlo simulations reported in Clark and McCracken (2013) demonstrate that (2.5) using the one-sized Gaussian critical values leads to decent sized test of the null hypothesis of equal finite-sample point forecast accuracy for nested models. It is also worth mentioning that Harvey et al. (1997) suggest a small-sample correction of (2.5). In the empirical analysis, I also apply the mentioned test. However, I find that both tests lead to similar qualitative conclusions.

The website link is: https://www.eia.gov/.

The website link is https://globalfinancialdata.com/.

Reliance on crude oil price data from 1947m1 until 1973m1 for econometric analysis is considered as somewhat controversial, see Alquist et al. (2013). The reason is that the price of crude oil varies little in this period, and exhibits a pattern resembling a step-function. As discussed in Hamilton (1983) this property is explained by the specific regulations imposed on the oil industry.

More specifically, Hamilton (2011) states “. deflating by a particular number, such as the CPI, introduces a new source of measurement error, which could lead to a deterioration in the forecasting performance. In any case, it is again quite possible that there are differences in the functional form of forecasts based on nominal instead of real prices...”.

The US CPI index is extracted from Michael W. McCracken’s website: https://research.stlouisfed.org/econ/mccracken/fred-databases/. The website contains a large set of historical macroeconomic time-series.

Inserting a trend term in the test equation is common when one tests commodities for unit root, see Narayan and Gupta (2015); Narayan and Liu (2015). It must also be mentioned that results using the augmented Dickey and Fuller (1981) test are similar to Phillips and Perron (1988).

Under the expanding window estimation approach, the time-period \(t+h\) point forecast of the variable of interest is generated conditional on information from time-period 1 to t. The estimation window is then rolled forward, and the point forecast for time-period \(t+2\) is produced using information from time-period 1 to \(t+1\) and so on.

The reader should note that nowhere in this study have there been any attempts made to optimize the performance of (2.1) relative to the end-of-month benchmark by modifying these settings, in recognition of the concern over data mining discussed in Rossi and Inoue (2012).

It is worth mentioning that all US recessions are also included in the recession periods for the OECD-based series.

In a number of studies, the authors report the out-of-sample \(R^{2}\). However, this quantity is simply one minus the TU.

Specifically, the max MSE-adj t-statistic variant of the test suggested in Hubrich and West (2010) is relied on.

The first term, Bias, is computed by average over point forecast errors over the out-of-sample period. The second term is computed by subtracting MSE from the first term squared.

Note that both variables in (4.1) are standardized prior to regression. Hence, the regression intercept, \(\alpha \), is omitted from (4.1).

Conlon et al. (2022) show that with regard to forecasting returns on the price of crude oil by conditioning on economic variables, results using monthly average prices are misleading. This is due to the fact that within-month averages of daily oil crude oil prices in calculating returns induce a bias, which, in turn, leads to false inference about the true extent of out-of-sample relative predictability. This does not occur when using end-of-month crude oil prices. Furthermore, once the end-of-month prices are relied on, then Conlon et al. (2022) demonstrate that contrary to the current literature, one does not find convincing evidence of relative out-of-sample predictability.

The first three variables are extracted from the website of Amit Goyal:: http://www.hec.unil.ch/agoyal/. The VIX index is extracted from Federal Reserve Economic Database: https://fred.stlouisfed.org/, and the remaining two variables are extracted from the website of Scott R. Baker, Nick Bloom and Steven J. Davis: https://www.policyuncertainty.com/about.html.

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Titel: Point forecasts of the price of crude oil: an attempt to “beat” the end-of-month random-walk benchmark
verfasst von: Nima Nonejad
Publikationsdatum: 03.05.2024
Verlag: Springer Berlin Heidelberg
Erschienen in: Empirical Economics
Print ISSN: 0377-7332
Elektronische ISSN: 1435-8921
DOI: https://doi.org/10.1007/s00181-024-02599-8

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